Question: Lisa, a child with strange requirements for her projects, is making a rectangular cardboard box with square bases. She wants the height of the box to be 3 units greater than the side of the square bases. What should the height be if she wants the surface area of the box to be at least 90 square units while using the least amount of cardboard?
Explanation: We let the side length of the base of the box be $x$, so the height of the box is $x+3$. The surface area of each box then is $2x^2 + 4(x)(x+3)$.  Therefore, we must have $$2x^2+4x(x+3) \ge 90.$$Expanding the product on the left and rearranging gives $ 2x^2+4x^2+12x-90 \ge 0$, so $6x^2+12x-90 \ge 0$.  Dividing by 6 gives  $x^2+2x-15 \ge 0$, so  $(x+5)(x-3) \ge 0$.  Therefore, we have $x \le -5 \text{ or } x\ge 3$. Since the dimensions of the box can't be negative, the least possible length of the side of the base is $3$. This makes the height of the box $3+3=\boxed{6}$.

Note that we also could have solved this problem with trial-and-error.  The surface area increases as the side length of the base of the box increases.  If we let this side length be 1, then the surface area is $2\cdot 1^2 + 4(1)(4) = 18$.  If we let it be 2, then the surface area is $2\cdot 2^2 + 4(2)(5) = 8 + 40 = 48$.  If we let it be 3, then the surface area is $2\cdot 3^2 + 4(3)(6) = 18+72 = 90$.  So, the smallest the base side length can be is 3, which means the smallest the height can be is $\boxed{6}$.